M is a nxn matrix where all entries are 1's.
My conjecture is that n - 1 eigenvalues are 0 and the nth eigenvalues is one.
If this is correct, how do I go about proving it?
first he subtracted the first row, from every other row. this changed all the 1's in the first column (except for the top entry) to -λ.
it also changed all the 1's in every other column to 0, except along the diagonal, where 1-λ becomes -λ.
then he sucessively added (it doesn't matter if you do it one at a time, or all at once) columns 2 through n, to the first column.
neither the row operation, nor the column operation changes the determinant, since no row or column was multiplied.